Riemannian and Sub-Riemannian Geodesic Flows
Mauricio Godoy Molina1Joint with E. Grong
Universidad de La Frontera (Temuco, Chile)
Potsdam, February 2017
1Partially funded by grant Anillo ACT 1415 PIA CONICYT (Chile)Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 1 / 24
Outline
1 Introduction: Geometry and restrictions
2 Geodesic Flows
3 Results
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 2 / 24
Philosophy: Restrictions in mechanics
In classical analytic mechanics we study systems of the form
Differential equation + restrictions.
Usually: restrictions on the position and/or velocity, but
restr. on the velocity can “hide” restr. on the position.
Definition (heuristic)Restrictions on the position (hidden or not) are called holonomic.
In what follows, we call non-holonomic to those restrictions on velocity nothiding restrictions on the position.
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 3 / 24
Hidden restrictions
In examples 1-3: a particle P moves in R3 with position −→r = −→r (t),velocity −→v = −→v (t) and under certain restrictions.
Example 1: −→v ⊥ −→r ⇐⇒P moves on the sphere of radius |−→r (0)| centered at −→0 .
In other words, −→v ⊥ −→r is holonomicsince it “hides a restriction on theposition.”
DefinitionThe restriction on −→v can beintegrated to a restriction on −→r .
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 4 / 24
Perturbing restrictionsExample 2: −→v ⊥
−→k ⇐⇒ −→v = a · −→ı + b · −→ ⇐⇒
P moves on a horizontal plane.
Holonomic again. But. . . what if wemodify the problem slightly?
Example 3: −→v ⊥(−→k − x−→
)⇐⇒ −→v = a · −→ı + b ·
(−→ + x−→k).
Non-holonomic! Moreover:
Theorem (Folklore)Small perturbations of restrictionsare non-holonomic.
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 5 / 24
Mathematical formulationPrevious fact: “Most” restrictions are non-holonomic.That’s why we want to understand them.
Space of configurations: Q, with dimQ = dof.Restrictions: −→v (t) ∈ spanX1, . . . ,Xk.
Example 1: Q = sphere, −→v (t) ∈ span∂θ, ∂ϕ
in spherical coordinates.
Example 2: Q = horizontal plane, −→v (t) ∈ span−→ı ,−→ .
Example 3: Q = R3, −→v (t) ∈ span−→ı ,−→ + x
−→k.
DefinitionThe flow of X ∈ Γ(Q) is the function Q × R→ Q given by
eτXp = solution ofγ(s) = X (γ(s))γ(0) = p
in time s = τ.
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 6 / 24
Detecting holonomy I
Flow of −→ı : eτ−→ı (x , y , z) = (x + τ, y , z).
Flow of −→ : eτ−→ (x , y , z) = (x , y + τ, z).
Flow of −→ + x−→k : eτ(−→ +x
−→k )(x , y , z) = (x , y + τ, z + τx).
Theorem (Frobenius, 1877)A restriction H = spanX1, . . . ,Xk is holonomic iff
ddτ
∣∣∣∣τ=0
e−√τY e−
√τXe
√τY e
√τXp ∈ H, ∀X ,Y ∈ H,∀p ∈ Q.
Explanation?
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 7 / 24
Detecting holonomy II
For simplicity: t =√τ . Suppose X and Y restrictions in H.
We follow X ,afterward Y ,we come back using X ,finally we come back using Y .
Is the velocity of this new curve a restriction?If yes, then H is holonomic.
DefinitionThe process described above is called commuting flows.
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 8 / 24
Detecting holonomy III
Example 2: Holonomic
ddτ
∣∣∣∣τ=0
e−√τ−→ e−
√τ−→ı e√τ−→ e√τ−→ı (x , y , z) =
ddτ
∣∣∣∣τ=0
(x , y , z)︸ ︷︷ ︸constant!
= (0, 0, 0) ∈ span−→ı ,−→ .
Example 3: Non-holonomic
ddτ
∣∣∣∣τ=0
e−√τ(−→ +x
−→k )e−
√τ−→ı e√τ(−→ +x
−→k )e√τ−→ı (x , y , z) =
ddτ
∣∣∣∣τ=0
(x , y , z + τ) = (0, 0, 1) =−→k /∈ span
−→ı ,−→ + x−→k.
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Outline
1 Introduction: Geometry and restrictions
2 Geodesic Flows
3 Results
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 10 / 24
Affine control systemsGiven H = spanX1, . . . ,Xk, k ≤ n, we have the control problem
x =k∑
i=1uiXi (x)⇐⇒ x ∈ H. (SAC)
Same question:Given x0, xT ∈ M, can we find u1, . . . , uk such that
x(0) = x0 y x(T ) = xT ?
BIG observation: The controllability of (SAC) is a consequence of Hbeing completely non-holonomic.
DefinitionH is completely non-holonomic if wecan obtain any velocity by commutingflows in H.
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Geodesics
Suppose that (SAC) is controllable.
x =k∑
i=1uiXi (x)⇐⇒ x ∈ H (SAC)
ProblemGiven x0, xT ∈ M, we want to find u = (u1, . . . , uk) such that
x(0) = x0, x(T ) = xT and
J(u) = minu
J(u) = minu
12
∫ (u1(t)2 + · · ·+ uk(t)2
)dt.
Usually: A curve γ with control u is called sR-geodesic if k < n andR-geodesic if k = n.
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Cometric and metric
Consider a bilinear non-negative tensor h∗ on T ∗M (a cometric) and
]h∗ : T ∗M → T ∗∗M = TM, p 7→ h∗(p, ·)
H = im ]h∗ are the restrictions on M endowed with the metric
h(]h∗p, ]h∗q) = h∗(p, q)
Fact(H,h) determines uniquely h∗. Besides ker ]h∗ = Ann H.
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Hamiltonian and flows
The Hamiltonian Hh associated to the metric h is simply
Hh(p) = 12h∗(p, p).
If ω is the canonical 2-form on T ∗M, then
DefinitionThe Hamiltonian vector field ~Hh is given by
dHh(X ) = ω(~Hh,X ), ∀X ∈ X(M).
If h is a (sub-)Riemannian metric, then et~Hh is the (sub-)Riemanniangeodesic flow.
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Outline
1 Introduction: Geometry and restrictions
2 Geodesic Flows
3 Results
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Motivating problem I
Theorem (Montgomery)Given a principal G-bundle G y M π→ N (+ technical hypotheses), thenthe normal sub-Riemannian geodesics on M defined by H = (ker dπ)⊥ aregiven by
γsr (t) = expr (tv) · expG(−tA(v)),
where A is the g-valued connection one form.
Idea:Compute Riemannian geodesics in M.Project down to N.“Horizontally lift” the curve to M.
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Horizontal lift
Given a submersion π : M → N and a vector v ∈ TxN, then at eachx ∈ π−1(x) there is a unique vector v ∈ TxM such that dxπ(v) = v . Thisis the horizontal lift of v . For a vector field X ∈ Γ(TN), define X by
X |x = X |x .
The horizontal lift γ of γ : [0,T ]→ N at2 x is the unique solution to
˙γ = ¯γ, γ(0) = x
2Obviously πx = γ(0)Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 17 / 24
Motivating problem II
Montgomery’s theorem can be used for some examples.Hopf fibration: U(1) y S2n+1 → CP1
Quaternionic Hopf fibration: Sp(1) y S4n+3 → HPn
Grassmannians: U(n) y Vn,k → Grn,k
BUT there are “true” fibrations: S7 → S15 → OP1 ∼= S8 (octonionicHopf). What to do when there is no group?
Remark (Ornea, Parton, Piccinni, Vuletescu 2013)The situation is worse than expected: ANY v.f. tangent to the leaves ofS7 → S15 → S8 has a zero.
Mauricio Godoy M. (UFRO) Geodesic Flows February, 2017 18 / 24
Taming metrics
Let (M,H,h) a sub-Riemannian manifold.A Riemannian metric g on M tames h if g|H = h.
“Natural” questionIs there a relation between the geodesics of g and the ones of h?
Let V = H⊥ with respect to g. Define v = g|V .
Technical toolThe following formula defines a conection (Bott)
∇XY =prH∇gprHX prHY + prV∇
gprVX prVY+
prH[prVX ,prHY ] + prV [prHX , prVY ].
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Key Lemma
Lemma (G., Grong)If ·, · denotes the Poisson bracket wrt ω, then
Hh,Hv = Hh,Hg = 0 ⇐⇒ ∇g = 0
If ΠM : T ∗M → M is the canonical projection, then
expsr : Ux ⊆ T ∗x M → M, expsr (x , tp) = (ΠM et~Hh)(p)
expr : Vx ⊆ TxM → M, expr (x , t]p) = (ΠM et~Hg)(p)
Consequence
expr (x , t]p) =(ΠM et~Hh et~Hv)(p) =
(ΠM et~Hv et~Hh)(p).
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Main result I
Theorem (G., Grong)(M, g) Riemannian + F tot. geodesic Riem. foliation of Mw/subbundle V. Define (M,H,h) where H = V⊥ and h = g|H. Then, forany x ∈ M and p ∈ T ∗M
expsr (x , tp) = expr (expr (x , t]p),−tprVPt]p) ,
where Pt is the parallel transport along expr (x , t]p)
Compare with Montgomery’s geodesics γsr (t) = expr (tv) · expG(−tA(v)).
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Main result II
If we assume that V is integrable, then we know more.
Theorem (G., Grong)Every curve of the form expsr (x , tp) is the horizontal lift of the projectionof the curve expr (x , t]p) iff(a) V is the orthogonal complement of H.(b) The leaves of the foliation of V are totally geodesic.
For the interested few: “Riemannian and Sub-Riemannian GeodesicFlows”. To appear J. Geom. Analysis (I guess this year)
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Examples
For principal bundles (+ technical conditions), the formula
expsr (x , tp) = expr (expr (x , t]p),−tprVPt]p)
coincides with Montgomery’s resultThe result can be applied to S7 → S15 → S8, but we have no explicitformulas yet (M.Sc. problem anyone?)If (M,H,h), where H = kerα for α ∈ Ω1(M) contact, then
∇g = 0 iff LZg = 0,
where Z is the Reeb vector field, g(Z ) = 1 and g|H = h
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